# Clausius Inequality application to engine

So I am aware of the Clausius inequality which states the following:

• ∮δQ/T≤0 around a cycle.

Now consider the following question:

Consider a real heat pump which in one cycle extracts heat Qc from the ground at temperature Tc and delivers heat Qh to a building at at temperature Th. Use the Clausius inequality to show that Qc < QhTc /Th.

• The solution to this answer is as follows:

Using the Clausius inequality:

Qc/Tc - Qh/Th < 0 --> Qc < QhTc /Th

My question is how did we know that Qc/Tc - Qh/Th < 0

as opposed to Qh/Th - Qc/Tc <0 which would obviously yield a different result.

• hint: $\delta Q$ is the energy the thermostat provides at temperature $T$ to the system satisfying the 1st law $dU = \delta Q + \delta W$. Thus $\delta Q$ is positive or negative whether the system absorbs or emits that amount of "heat", resp. – hyportnex Apr 13 '18 at 22:00